Finite Difference Methods for Caputo-Hadamard Fractional Differential Equations
Finite Difference Methods for Caputo-Hadamard Fractional Differential Equations
复制标题
Caputo–Hadamard 分数阶微分方程的有限差分法
DOI：
10.1007/s00009-020-01605-4
复制
发表时间：
2020-10-27
影响因子：
1.1
通讯作者：
Li, Zhiqiang
中科院分区：
文献类型：
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作者：
Gohar, Madiha;Li, Changpin;Li, Zhiqiang
本文研究了含Caputo-Hadamard导数的分数阶微分方程的有限差分方法。首先，光滑性的解决方案进行了研究。通过逼近相应的等价沃尔泰拉积分方程，提出了求解Caputo-Hadamard分数阶常微分方程的分数阶矩形、Llog、1插值和修正的预估校正方法.证明了所导出方法的稳定性和误差估计。然后，我们研究了含Caputo-Hadamard导数的分数阶偏微分方程的有限差分方法。利用构造的L1格式逼近时间分数阶导数，得到了一个半离散差分格式。文中还给出了算法的稳定性和收敛性分析。在空间方向上，利用标准二阶差分格式建立了全离散格式。稳定性和误差估计。数值实验验证了理论结果。
In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo-Hadamard derivatives. First, smoothness properties of the solution are investigated. The fractional rectangular, Llog,1 interpolation, and modified predictor-corrector methods for Caputo-Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo-Hadamard derivative. By applying the constructed L1 scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.